Abstract
In this article, we investigate the Ulam-Hyers stability of C*-ternary algebra n-homomorphisms for the functional equation:
in C*-ternary algebras.
2000 Mathematics Subject Classification: Primary 39B82; 46B03; 47Jxx.
Journal of Inequalities and Applications volume 2011, Article number: 30 (2011)
In this article, we investigate the Ulam-Hyers stability of C*-ternary algebra n-homomorphisms for the functional equation:
in C*-ternary algebras.
2000 Mathematics Subject Classification: Primary 39B82; 46B03; 47Jxx.
Ternary algebraic operations were considered in the nineteenth century by several mathematicians, such as Cayley [1] who introduced the notion of cubic matrix, which, in turn, was generalized by Kapranov et al. [2]. The simplest example of such nontrivial ternary operation is given by the following composition rule:
Ternary structures and their generalization, the so-called n-array structures, raise certain hopes in view of their applications in physics. Some significant physical applications are as follows (see [3]):
The algebra of nonions generated by two matrices
was introduced by Sylvester as a ternary analog of Hamilton's quaternions [4].
The quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechnics is based on such structures [5].
There are also some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and Yang-Baxter equation [4, 6].
A C*-ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) α [x, y, z] of A^{3} into A, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies ||[x, y, z]|| ≤ ||x|| · ||y|| · ||z|| and ||[x, x, x]|| = ||x||^{3} (see [7, 8]). Every left Hilbert C*-module is a C*-ternary algebra via the ternary product [x, y, z] := 〈x, y〉 z.
If a C*-ternary algebra (A,[·, ·, ·]) has an identity, i.e., an element e ∈ A such that x = [x, e, e] = [e, e, x] for all x ∈ A, then it is customary to verify that A, endowed with x ∘ y := [x, e, y] and x* := [e, x, e], is a unital C*-algebra. Conversely, if (A, ∘) is a unital C*-algebra, then [x, y, z] := x ∘ y* ∘ z makes A into a C*-ternary algebra.
Let A and B be C*-ternary algebras. A ℂ-linear mapping H : A → B is called a C*-ternary algebra homomorphism if
for all x, y, z ∈ A.
Definition. Let A and B be C*-ternary algebras. A multilinear mapping H : A^{n} → B over ℂ is called a C*-ternary algebra n-homomorphism if it satisfies
for all x_{1}, y_{1}, z_{1}, · · ·, x_{ n } , y_{ n } , z_{ n } ∈ A.
In 1940, Ulam [9] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms:
We are given a group G and a metric group G' with metric ρ(·, ·). Given ε > 0, does there exist a δ > 0 such that if f : G → G' satisffies ρ(f(xy), f(x) f(y)) < δ for all x, y ∈ G, then a homomorphism h : G → G' exists with ρ(f(x), h(x)) < ε for all x ∈ G ?
In 1941, Hyers [10] gave the first partial solution to Ulam's question for the case of approximate additive mappings under the assumption that G_{1} and G_{2} are Banach spaces. Then, Aoki [11] and Bourgin [12] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [13] generalized the theorem of Hyers [10] by considering the stability problem with unbounded Cauchy differences. In 1991, Gajda [14], following the same approach as that by Rassias [13], gave an affirmative solution to this question for p > 1. It was shown by Gajda [14] as well as by Rassias and Šemrl [15], that one cannot prove a Rassias-type theorem when p = 1. Găvruta [16] obtained the generalized result of Rassias's theorem which allows the Cauchy difference to be controlled by a general unbounded function. During the last two decades, a number of articles and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings, for example, Cauchy-Jensen mappings, k-additive mappings, invariant means, multiplicative mappings, bounded n th differences, convex functions, generalized orthogonality mappings, Euler-Lagrange functional equations, differential equations, and Navier-Stokes equations. The instability of characteristic flows of solutions of partial differential equations is related to the Ulam's stability of functional equations [17]-[24]. On the other hand, the authors [25], Park [20] and Kim [26] have contributed studies in respect of the stability problem of ternary homomorphisms and ternary derivations.
Let X and Y be real or complex vector spaces and n ≥ 2 an integer. For a mapping f : X^{n} → Y, consider the functional equation:
The above functional equation is rewritten as
where
We solve the general problem in vector spaces for the n-additive mappings satisfying (2.1).
Theorem 2.1. A mapping f : X^{n} → Y satisffies the equation (2.1) if and only if the mapping f is n-additive.
Proof. Assume that f satisfies (2.1). Letting x_{1} = y_{1} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we get f(0, · · ·, 0) = 0. Letting y_{1} = x_{2} = y_{2} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we have
for all x_{1} ∈ X. Similarly, we get
for all x_{2}, · · ·, x_{ n } ∈ X. Setting y_{1} = y_{2} = 0 and x_{3} = y_{3} = · · · = x_{ n } = y_{ n } = 0 in (2.2), we have
for all x_{1}, x_{2} ∈ X. Similarly, we get f(x_{1}, 0, x_{3}, 0, · · ·, 0) = · · · = f(0, · · ·, 0, x_{n-1}, x_{ n } ) = 0 for all x_{1}, · · ·, x_{ n } ∈ X.
Continuing this process, we obtain that f(x_{1}, · · ·, x_{ n } ) = 0 for all x_{1}, · · ·, x_{ n } ∈ X with x_{ i } = 0 for some i = 1, · · ·, n. Letting y_{2} = · · · = y_{ n } = 0 in (2.2), we get the additivity in the first variable. Similarly, the additivities in the remaining variables hold.
The converse is obvious. □
We investigate the generalized Ulam's stability in C*-ternary algebras for the n-additive mappings satisfying (2.1).
Lemma 2.2. Let X and Y be complex vector spaces and let f : X^{n} → Y be a n-additive mapping such that
for alland all x_{1}, · · ·, x_{ n } ∈ X, then f is n-linear over ℂ.
Proof. Since f is n-additive, we get for all x_{1}, · · ·, x_{ n } ∈ X. Now let and M be an integer greater than 2(|σ_{1}| + · · · + |σ_{ n } |). Since , there is such that . Now
for some . Thus, we have
for all x_{1}, · · ·, x_{ n } ∈ X. Hence, the mapping f : X^{n} → Y is n-linear over ℂ. □
Using the above lemma, one can obtain the following result.
Theorem 2.3. Let X and Y be complex vector spaces and let f : X^{n} → Y be a mapping such that
for alland all x_{1,1}, x_{2,1}, · · ·, x_{1,n}, x_{2,n}∈ X. Then, f is n-linear over ℂ.
Proof. Letting λ_{1} = · · · = λ_{ n } = 1, by Theorem 1.1, f is n-additive. Letting x_{2,1} = · · · = x_{2,n}= 0 in (2.3), we get f(λ_{1}x_{1}, · · ·, λ_{ n }x_{ n } ) = λ_{1} · · · λ_{ n }f(x_{1}, · · ·, x_{ n } ) for all and all x_{1}, · · ·, x_{ n } ∈ X. Hence, by Lemma 2.2, the mapping f is n-linear over ℂ. □
From now on, assume that A is a C*-ternary algebra with norm || · || _{ A } and that B is a C*-ternary algebra with norm || · ||_{ B }.
For a given mapping f : A^{n} → B, we define
for all and all x_{1,1}, x_{2,1}, · · ·, x_{1,n}, x_{2,n}∈ A.
We prove the generalized Ulam-Hyers stability of homomorphisms in C*-ternary algebras for the functional equation
Theorem 2.4. Let p_{1}, ···, p_{ n } ∈ (0, ∞) withand θ ∈ (0, ∞), and let f : A^{n} → B be a mapping such that
and
for alland all x_{1}, y_{1}, z_{1}, · · ·, x_{ n } , y_{ n } , z_{ n } ∈ A. Then, there exists a unique C*-ternary algebra n-homomorphism H : A^{n} → B such that
for all x_{1}, · · ·, x_{ n } ∈ A.
Proof. Letting λ_{1} = · · · = λ_{ n } = 1, y_{1} = x_{1}, · · ·, y_{ n } = x_{ n } in (2.4), we gain
for all x_{1}, · · ·, x_{ n } ∈ A. Thus, we have
for all x_{1}, · · ·, x_{ n } ∈ A and all j ∈ ℕ. For given integer l, m(0 ≤ l < m), we obtain that
for all x_{1}, · · ·, x_{ n } ∈ A. Since , the sequence
is a Cauchy sequence for all x_{1}, ···, x_{ n } ∈ A. Since B is complete, the sequence converges for all x_{1}, · · ·, x_{ n } ∈ A. Define H : A^{n} → B by
for all x_{1}, · · ·, x_{ n } ∈ A. Letting l = 0 and taking m → ∞ in (2.8), one can obtain the inequality (2.6). By (2.4), we see that
for all x_{1}, y_{1}, · · ·, x_{ n } , y_{ n } ∈ A and all s. Since , letting s → ∞ in the above inequality, H satisfies (2.1). By Theorem 2.1, H is n-additive.
Letting y_{1} = x_{1}, · · ·, y_{ n } = x_{ n } in (2.4), we gain
for all and all x_{1}, · · ·, x_{ n } ∈ A. Thus we have
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Hence, we get
for all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ, and one can show that
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Hence,
for all , all x_{1}, · · ·, x_{ n } ∈ A and all m ∈ ℕ. Since , we have
as m → ∞ for all and all x_{1}, · · ·, x_{ n } ∈ A. Hence
for all and all x_{1}, ···, x_{ n } ∈ A. From Lemma 2.2, the mapping H : A^{n} → B is n-linear over ℂ. It follows from (2.5) that
for all x_{1}, y_{1}, z_{1}, ···, x_{ n } , y_{ n } , z_{ n } ∈ A. So
for all x_{1}, y_{1}, z_{1}, ···, x_{ n } , y_{ n } , z_{ n } ∈ A.
Now, let T : A^{n} → B be another n-additive mapping satisfying (2.6). Then, we have
which tends to zero as m → ∞ for all x_{1}, ···, x_{ n } ∈ A. Hence, we can conclude that H(x_{1}, ···, x_{ n } ) = T(x_{1}, ···, x_{ n } ) for all x_{1}, ···, x_{ n } ∈ A. This proves the uniqueness of H.
Thus, the mapping H : A→B is a unique C*-ternary algebra n-homomorphism satisfying (2.6). □
Letting p_{1} = ··· = p_{ n } = 0 and θ = ε in Theorem 2.4, we obtain the Ulam-Hyers stability for the n-additive functional equation (2.1).
Corollary 2.5. Let ε ∈ (0, ∞) and let f : A^{n} → B be a mapping satisfying
and
for alland all x_{1}, y_{1}, z_{1} ···, x_{ n } , y_{ n } , z_{ n } ∈ A. Then, there exists a unique C*-ternary algebra n-homomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Example 2.6. We present the following counterexample modiffied by the well-known counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let.
Deffine a function f :ℝ ^{n} → ℝ by
for all x_{1}, ···, x_{ n } ∈ ℝ, where ϕ_{ μ } : ℝ ^{n} → ℝ is the function given by
for all x_{1}, ···, x_{ n } ∈ ℝ. Deffine another function g : ℝ → ℝ by
for all x ∈ ℝ.
For all m ∈ ℕ and all, we assert that
It was proved in[14]that
for all x, y ∈ ℝ, that is, (2.9) holds for m = 1. For a ffixed k ∈ ℕ, assume that (2.9) holds for m = k. Then, we have
for all, that is, (2.9) holds for m = k + 1. Hence, (2.9) holds for all m ∈ ℝ.
Note that
for all x_{1}, ···, x_{ n }∈ ℝ. By the inequality (2.9) and the above equality, we see that
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n } ∈ ℝ. However, we observe from[14]that
and so
Thus,
where h: ℝ^{n} → ℝ is the function given by
for all x_{1}, ···, x_{ n }∈ ℝ. Hence, the function f is a counterexample for the singular caseof Theorem 2.4.
Theorem 2.7. Let p ∈ (0,n) and θ ∈ (0, ∞), and let f : A^{n} → B be a mapping such that
and
for alland all x_{1}, y_{1}, z_{1} ···, x_{ n }, y_{ n }, z_{ n }∈ A. Then, there exists a unique C*-ternary algebra n-homomorphism H : A^{n}→B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to the proof of Theorem 2.4. □
Example 2.8. We present the following counterexample modiffied by the well-known counterexample of Z. Gajda[14]for the functional equation (2.1). Set θ > 0 and let
Let f : ℝ^{n} → ℝ and g : ℝ → ℝ be the same as in Example 2.6. By the same argument as in Example 2.6, for all m ∈ ℕ and all, one can obtain that g satisffies the inequality:
By the equality (2.10) and the above inequality, we see that
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ. For each x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ, let M(x_{1}, y_{1}, ···, x_{ n } , y_{ n }) := max{|x_{1}|, |y_{1}|, ···, |x_{ n } |, |y_{ n } |}. We have
for all x_{1}, y_{1}, ···, x_{ n } , y_{ n }∈ ℝ. Thus we have
for all x_{1}, y_{1}, ···, x_{ n }, y_{ n }∈ ℝ. By the same reason as for Example 2.6, the function f is a counterexample for the singular case p = n of Theorem 2.7.
Theorem 2.9. Let p_{1}, ···, p_{ n } ∈ (0, ∞) with, s ∈ (0, n) and θ, η ∈ (0, ∞), and let f: A^{n} → B be a mapping such that
and
for alland all x_{1,1}, x_{2,1}, x_{3,1}, ···, x_{1, n}, x_{2, n}, x_{3, n}∈ A. Then, there exists a unique C*-ternary algebra n-homomorphism H: A^{n}→ B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to the proof of Theorem 2.4. □
Theorem 2.10. Let p_{1}, ···, p_{ n } ∈ (0, ∞) withand θ ∈ (0, ∞), and let f : A^{n} → B be a mapping satisfying (2.4) (2.5). Then, there exists a unique C*-ternary algebra n-homomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. It follows from (2.7) that
for all x_{1}, ···, x_{ n } ∈ A. Hence,
for all nonnegative integers m and l with m > l and all x_{1}, ···, x_{ n } ∈ A. It follows from (2.16) that the sequence is a Cauchy sequence for all x_{1}, ···, x_{ n } ∈ A. Since B is complete, the sequence converges. Hence, one can define the mapping H : A^{n} → B by for all x_{1}, ···, x_{ n } ∈ A. Moreover, letting l = 0 and passing the limit m → ∞ in (2.16), we get (2.15).
The remainder of the proof is similar to the proof of Theorem 2.4. □
Theorem 2.11. Let p ∈ (3n, ∞) and θ ∈ (0, ∞), and let f: A^{n} → B be a mapping satisfying (2.11) (2.7), and f(0, ···, 0) = 0. Then, there exists a unique C*-ternary algebra n-homomorphism H : A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to that of Theorem 2.10. □
Theorem 2.12. Let p_{1}, ···, p_{ n } ∈ (0, ∞) with, s ∈ (n, ∞) and θ, η ∈ (0, ∞), and let f: A^{n} → B be a mapping such that (2.13), (2.14), and f(0, ···, 0) = 0. Then, there exists a unique C*-ternary algebra n-homomorphism H: A^{n} → B such that
for all x_{1}, ···, x_{ n } ∈ A.
Proof. The proof is similar to that of Theorem 2.10.
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The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.
The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Park, WG., Bae, JH. Generalized ulam-hyers stability of C*-Ternary algebra n-Homomorphisms for a functional equation. J Inequal Appl 2011, 30 (2011). https://doi.org/10.1186/1029-242X-2011-30
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DOI: https://doi.org/10.1186/1029-242X-2011-30